Say a finite dimensional algebra $A$ satisfies the right UC-condition if there exists an indecomposable projective module $P$ of $A$ such that $\operatorname{injdim}(\operatorname{top}(P))=1$ and $P$ has a minimal injective coresolution $(I_i)$ such that $$\dim I_0 \leq 2 (\dim I_1 - \dim I_2 + \dim I_3 - \dots).$$ Dually say that $A$ satisfies the left UC-condition if the dual statement is true (or equivalently, if the opposite algebra of $A$ satisfies the right UC-condition). And $A$ satisfies the UC-condition if it satisfies the left and right version.
For $A$ being the incidence algebra of a finite lattice $L$ this condition is equivalent to the Union-closed sets conjecture, see A homological algebra approach to the Union-closed sets conjecture . But already for incidence algebras of general posets the UC-condition is not always true anymore.
Question: Can one expect that there is a more general class of finite dimensional algebras containing the incidence algebras of lattice such that the UC-condition is always true?
(one might ask: When does a finite dimensional algebra behave like "a lattice"?)
Of course my generalisation of the conjecture to a condition for general algebras might not be the best. For example for lattices being join-irreducible for a point $x$ is equivalent to $\operatorname{injdim}(\operatorname{top}(e_xA))=1$ and to $\dim(\operatorname{Ext}_A^1(A/J,\operatorname{top}(e_x A)))=1$, but for general algebras those two conditions are not equivalent in general and I choose the condition $\operatorname{injdim}(\operatorname{top}(e_xA))=1$ over $\dim(\operatorname{Ext}_A^1(A/J,\operatorname{top}(e_x A)))=1$, since it is more "homological". Maybe someone has a suggestion for other generalisations.
Finite dimensional algebras do in general not even have an indecomposable projective module $P$ with $\operatorname{injdim}(\operatorname{top}(P))=1$ but such modules exist when the quiver of $A$ is acyclic.
One can for example look at the UC-condition now for other combinatorial objects such as Dyck paths which are in natural bijection to linear Nakayama algebras (see for examaple https://arxiv.org/abs/1811.05846 ). This motives the following question:
Question: Which Dyck paths satsify the UC-condition?
(see http://www.findstat.org/StatisticsDatabase/St001594 for the statistic)
It seems very hard to check the condition in general but for some special subclasses there seem to be nice patterns. Here one example for bouncing Dyck paths, that are in natural bijection to integer compositions (see for example What are the periodic Dyck paths? for the definition of bouncing Dyck paths):
For $n \geq 3$ the number of bouncing Dyck paths that satisfy the right UC-condition starts with 1,2,5,11,24,51,107,222,457 and might be given by https://oeis.org/A027934 and the sequence of bouncing Dyck paths that satisfy the two-sided UC-condition starts with 1,4,9,21,46,99,209 for $n=4,...,10$ and might be given by https://oeis.org/A027973 .
